The minimal interaction induced by the translation subgroup has a gap and possible relates to the strong fundamental interaction.

The paper investigates the low-symmetric state of the compensating field of the distortion tensor and proves that there is a gap in this state. It is shown that the distortion tensor is the compensating field of the minimal interaction induced by the translation subgroup. On the example of electron pairing in a Cooper pair it was proved that the distortion tensor is responsible for the electron-phonon interaction. In this paper, for the first time, an exact wave solution for sound pressure in a continuous medium is obtained from the equations of state for the distortion tensor. It is shown that the sound is described as "massive" wave of the distortion tensor, the spectrum of which has the minimal frequency, which corresponds to a gap. The presence of a gap in the low-symmetric state gives grounds to believe that the distortion tensor, as a compensating interaction field, describes a strong fundamental interaction. As it is known, the description of the gap in the strong fundamental interaction is declared a Millennium problem by the Clay Mathematical Institute (CMI).

The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames

In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as π(Γ)ψ , where π is a unitary representation of a wavelet group and Γ is the abstract pseudo-lattice Γ. We prove a sufficent condition in order that a Parseval frame π(Γ) ψ can be dilated to an orthonormal basis of the form τ (Γ) ψ, where τ is a super-representation of π. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications.

Affine descents and the Steinberg torus

International audience Let $W \ltimes L$ be an irreducible affine Weyl group with Coxeter complex $\Sigma$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $\Sigma$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $\gamma$-vector, and hence, symmetric and unimodal coefficients. In the classical cases, we also provide expansions, identities, and generating functions for the $h$-polynomials of Steinberg tori. Nous considérons un groupe de Weyl affine irréductible $W \ltimes L$ avec complexe de Coxeter $\Sigma$, où $W$ désigne le groupe de Weyl fini associé et $L$ le sous-groupe des translations. Le tore de Steinberg est le complexe cellulaire Booléen obtenu comme le quotient de $\Sigma$ par $L$. Nous montrons que les $h$-polynômes, ordinaires et de drapeaux, du tore de Steinberg (sans la face vide) sont des fonctions génératrices sur $W$ pour une statistique de type descente, étudiée en premier lieu par Cellini. Nous montrons également qu'un $h$-polynôme ordinaire possède un $\gamma$-vecteur positif, et par conséquent, a des coefficients symétriques et unimodaux. Dans les cas classiques, nous donnons également des développements, des identités et des fonctions génératrices pour les $h$-polynômes des tores de Steinberg.

GAUGE TRANSFORMATIONS, BRST COHOMOLOGY AND WIGNER'S LITTLE GROUP

We discuss the (dual-)gauge transformations and BRST cohomology for the two (1+1)-dimensional (2D) free Abelian one-form and four (3+1)-dimensional (4D) free Abelian two-form gauge theories by exploiting the (co-)BRST symmetries (and their corresponding generators) for the Lagrangian densities of these theories. For the 4D free two-form gauge theory, we show that the changes on the antisymmetric polarization tensor eμν(k) due to (i) the (dual-)gauge transformations corresponding to the internal symmetry group, and (ii) the translation subgroup T(2) of the Wigner's little group, are connected with each other for the specific relationships among the parameters of these transformation groups. In the language of BRST cohomology defined with respect to the conserved and nilpotent (co-)BRST charges, the (dual-)gauge transformed states turn out to be the sum of the original state and the (co-)BRST exact states. We comment on (i) the quasitopological nature of the 4D free two-form gauge theory from the degrees of freedom count on eμν(k), and (ii) the Wigner's little group and the BRST cohomology for the 2D one-form gauge theory vis-à-vis our analysis for the 4D two-form gauge theory.

WIGNER'S LITTLE GROUP AND BRST COHOMOLOGY FOR ONE-FORM ABELIAN GAUGE THEORY

We discuss the (dual-)gauge transformations for the gauge-fixed Lagrangian density and establish their intimate connection with the translation subgroup T(2) of Wigner's little group for the free one-form Abelian gauge theory in four (3+1)-dimensions (4D) of space–time. Though the relationship between the usual gauge transformation for the Abelian massless gauge field and T(2) subgroup of the little group is quite well known, such a connection between the dual-gauge transformation and the little group is a new observation. The above connections are further elaborated and demonstrated in the framework of Becchi–Rouet–Stora–Tyutin (BRST) cohomology defined in the quantum Hilbert space of states where the Hodge decomposition theorem (HDT) plays a very decisive role.

Corrections to ‘Invariant measures for higher-rank hyperbolic abelian actions’

The proofs of Theorems 5.1 and 7.1 of [2] contain a gap. We will show below how to close it under some suitable additional assumptions in these theorems and their corollaries. We will assume the notation of [2] throughout. In particular, $\mu$ is a measure invariant and ergodic under an $R^k$-action $\alpha$. Let us first explain the gap. Both theorems are proved by establishing a dichotomy for the conditional measures of $\mu$ along the intersection of suitable stable manifolds. They were either atomic or invariant under suitable translation or unipotent subgroups $U$. Atomicity eventually led to zero entropy. Invariance of the conditional measures showed invariance of $\mu$ under $U$. We then claimed that $\mu$ was algebraic using, respectively, unique ergodicity of the translation subgroup on a rational subtorus or Ratner's theorem (cf. [2, Lemma 5.7]). This conclusion, however, only holds for the $U$-ergodic components of $\mu$ which may not equal $\mu$. In fact, in the toral case, the $R^k$-action may have a zero-entropy factor such that the conditional measures along the fibers are Haar measures along a foliation by rational subtori. Since invariant measures with zero entropy have not been classified, we cannot conclude algebraicity of the total measure $\mu$ at this time. In the toral case, the existence of zero entropy factors turns out to be precisely the obstruction to our methods. The case of Weyl chamber flows is somewhat different as the ‘Haar’ direction of the measure may not be integrable. In this case, we need to use additional information coming from the semisimplicity of the ambient Lie group to arrive at the versions of Theorem 7.1 presented below.